# Math Help - Series Divergence

1. ## Series Divergence

I'm finding this question a bit difficult

Let $s_n$ denote the $n$th partial sum of the series:

$1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+...$

1) Show that $s_n>\sqrt{n}$ whenever $n>1$

2) Hence explain why the series diverges.

All I notice from this is that the series is:

$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}$

And that this is a p-series which must obviously diverge

2. 1. $n,i\in Z^{+}~,~n>i\rightarrow \displaystyle\sqrt{\frac{n}{i}}>1\rightarrow \sqrt{n}\sum_{i=1}\frac{1}{\sqrt{i}}>n$

$\rightarrow \displaystyle S_n=\sum_{i=1}\frac{1}{\sqrt{i}}>\sqrt{n}$

2. $\displaystyle S_n=\sum_{i=1}\frac{1}{\sqrt{i}}>\sum_{i=1}\frac{1 }{i}$ , so series diverges.